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Metamath Proof Explorer

Theorem chjoi

Description: The join of a closed subspace and its orthocomplement. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

Ref Expression
Hypothesis ch0le.1 ⊢ 𝐴 ∈ Cℋ
Assertion chjoi ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ

Proof

Step Hyp Ref Expression
1 ch0le.1 ⊢ 𝐴 ∈ Cℋ
2 1 chssii ⊢ 𝐴 ⊆ ℋ
3 ssjo ⊢ ( 𝐴 ⊆ ℋ → ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ )
4 2 3 ax-mp ⊢ ( 𝐴 ∨ℋ ( ⊥ ‘ 𝐴 ) ) = ℋ